We used the maximum likelihood D-optimal selection technique based on convex relaxation found in [1]} in order to choose sensors to try to reconstruct only the 3rd and 4th principal components of the shock-mixing layer snapshots.
That is, if \({\mathbf {U}} = \begin{bmatrix} {\mathbf {u}}_1 & {\mathbf {u}}_2 & \cdots \end{bmatrix}\)  is the matrix of principal components, we model the state as a linear combination of \({\mathbf {u}}_3\)  and \({\mathbf {u}}_4\)  together with isotropic Gaussian noise.
We try to find the sensors so that the correct coefficients on \({\mathbf {u}}_3\)  and \({\mathbf {u}}_4\)  can be recovered with high confidence from the measurements.
The rationale for doing so is the fact that these two components are sufficient to nonlinearly reconstruct the state of the system if they can be measured.
As in Section REF  above, we use a direct Python translation of a MATLAB code written by S. Joshi and
S. Boyd, which may be found at https://web.stanford.edu/~boyd/papers/matlab/sensor_selection/.
